Can I wear my Branch Chief insignia on my CG Unit Ball Cap?

You may wear your metal office collar insignia (gold oak leaf, with red 'A') on your CG Unit ball cap or your Auxiliary ball cap.

From the Auxiliary Manual, COMDINST M16790.1 (series), Sec. D.3.c. Coast Guard Unit Ball Cap:
The principal consideration about Auxiliarists‘ wear of a unit ball cap is that they must be authorized to wear it by the unit Commanding Officer...The Coast Guard Auxiliary garrison cap device is not displayed on this ball cap. Auxiliarists must wear either the metal office collar insignia or the member collar insignia (embroidered/sew-on versions are not an option).